No, no please let us be more precise to avoid confusion. The Mycenaeans enter on the scene in the 15th century bc with their attack on Crete, the Hittites at least 2 centuries earlier. They do not collapse in 1300-1200 bc, that century saw the construction of some of the most iconic monuments of both civilizations such as the Tomb of Agamemnon, the Lions' gate and Tyrins' gallery in Mycenaean Greece and the cyclopean galleries at Hattusash along with many iconic stone reliefs. The Mycenaeans fell slightly before the Hittites around the end of that century, and the theory of a "Dorian invasion" is pretty outdated, there is no evidence for an external invasion, and both Mycenaeans and Dorians were Greek speakers. The Hittites fell around 1180 bc, a few decades after the Mycenaeans, and it was not because of the Ahhyawa, who were likely the Mycenaeans by the way, but likely because of the civil war that had been going on for 70 years along with a massive drought that had hit all the Mediterranean, the Kaska invaders in the North might have also played a part, but the last enemies that the Hittites faced were an unnamed force coming from Alashiya (Cyprus) and the Lukka (Lycians), the Lycians had been known for several centuries by the Hittites. Since you named the Ahhyawa, they were mentioned by the Hittite texts since the early 14th century bc, they were at one point ruled by a king and had Greek names, they also were located West of Western Anatolia, likely somewhere in Greece, there is no doubt anymore that they were Mycenaean Greeks, Achaeans, and while they sometimes supported the West Anatolian kings against the Hittites they were never mentioned as a major threat by the Hittites. As for the Phrygians, they are not mentioned even once by the Hittites and likely migrated to central Anatolia sometime after the collapse of the Hittite empire, not before.Mycenaeans and Hittites arrive on the scene at about the same time (1700-1600 BCE), and their empires also collapse about the same time (1300-1200 BCE). (The Hyksos, a Semitic people, also invaded the Nile Delta around 1650 BCE.) The first was attacked and destroyed by the Dorians and the second by the Phrygians and the "Ahhiyawa" (Achaeans?), part of the overall movement of peoples called "the Sea Peoples". Both periods are denoted by mass migrations, warfare, raiding/piracy, droughts/climate change, famine, plague(?), etc.
Historically the steppes have served as a migration engine. Any drought, I assume, would hit the drier eastern steppes first, impelling people and their herds westward in search of water and greener grass. Wet cycles would cause populations (people and herds) to expand, with thus severe dry cycles putting everybody in motion, like balls on a billiard table. Agricultural civilizations, which used irrigation, dams, dikes, and canals to move water from rivers to fields were better able to withstand drought cycles in place, unless attacked by "barbarians" (in search of surplus grain) and brought down.